The purpose of this project is to develop a model that will predict the popularity of Spotify songs.
Spotify is an audio streaming service that was founded on April 23, 2006 in Stockholm, Sweden by Daniel Ek and Martin Lorentzon. This service includes both music and podcasts with its main usage being for music. Spotify has millions of songs with collections of music providing both a free and premium option for its subscribers.
knitr::include_graphics("images/spotifylogo.png")
Music brings the world together but have you ever wondered what makes some of the songs you love so great? The research question I am proposing is “What characteristics in a song can be used to determine how popular it can get?” The main motive is to find key factors that affects virality and trends that could be followed. As an avid Spotify user, I catch myself playing certain songs on repeat and knowing all the words to some of the greatest hits. It would be interesting to see if there are patterns with these songs and if they are overlapping attributes. By creating a model, this will allow Spotify users to figure out common patterns in the top Spotify songs and what makes them great hits and provide insight to artists about what they can produce in hopes of making a popular song.
knitr::include_graphics("images/taylorswift.png")
The data set being used is from Kaggle and it is called “Top Spotify Songs From 2010-2019 By Year” created by Leonardo Henrique. With this data, I can analyze popular songs and create a model that find potential trends.
Let’s discuss the plan for how to tackle this project. Fortunately,
this data has been cleaned and there are no missing values. First, we
will load the data and then perform exploratory data analysis. We will
use all the predictors given and have pop be the response
variable. pop indicates the value of how popular the song
is with 0 being not popular and 100 being the most popular, which is a
categorical variable. Linear Regression, Ridge Regression, K-Nearest
Neighbors, Elastic Net, Random Forest, Support Vector Machine, and
Boosted Trees. I will be used as models for the training data. From
there, we will use it on the testing data to see how well our model
actually did.
Before we start building models, we will begin with a general analysis of our data and figure out what we are working with.
# Loading all necessary packages
library(readr)
library(janitor)
library(dplyr)
library(tidymodels)
library(kknn)
library(glmnet)
library(corrr)
library(corrplot)
library(tidyverse)
library(finalfit)
library(kableExtra)
library(kernlab)
library(xgboost)
# Loading the data
spotify <- read_csv("top10s.csv", show_col_types = FALSE)
# Cleaning predictor names
spotify <- clean_names(spotify)
set.seed(100)
# Seeing first few rows of data set
head(spotify)
dim(spotify)
## [1] 603 15
There are 603 observations and 15 variables. This is a good amount of
observations and the amount of predictors we have is also good. As for
our predictors, we use bpm, nrgy,
dnce, d_b, live,
val, dur, acous, and
spch. pop will be our response.
# Missing values (none)
spotify %>%
missing_plot()
Luckily, there are no missing observations in our data set and we can use all the predictors for our model! As for tidying the data, the data set is pretty efficient and all components can and will be used.
After looking all the possible predictors, I have decided to keep all
the numeric variables since they provide good insight on the response
variable, pop. I decided not to use top_genre
as a predictor because the majority of the songs fall under some
category of pop (this will be shown in the Visual EDA portion). So, here
are the variables that will be used:
bpm: Beats Per Minute - the tempo of the songnrgy: Energy - the energy of the song, the higher the
value, the more energetic the songdnce: Danceability - the higher the value, the easier
it is to dance to the songd_b: Loudness - the higher the value, the louder the
songlive: Liveness - the higher the value, the more likely
the song is a live recordingval: Valence - the higher the value, the more positive
mood for the songdur: Length - the duration of the songacous: Acousticness - the higher the value, the more
acoustic the song isspch: Speechiness - the higher the value, the more
spoken word the song containspop: Popularity - the higher the value, the more
popular the song isLet’s check out a correlation heat map to see if there is a relationship between the variables.
spotify_numeric <- spotify %>%
select_if(is.numeric) %>%
select(-x1) %>%
select(-year)
spotify_cor <- cor(spotify_numeric)
spotify_cor_plt <- corrplot(spotify_cor, col = COL2("BrBG"))
When looking at the correlation plot, I thought that there would be
more correlation between the variables. The ones that have a stronger
correlation and the ones that were not correlated at strongly made
sense. As shown above, there is a correlation between d_b
and nrgy which makes sense because the louder the music, it
would typically also mean that the energy of the song is higher. In
addition, there is also a correlation between val and
dnce. This is expected because most songs people dance too
are typically more positive songs that are more upbeat. On the other
hand, there is a negative correlation between ngry and
acous which is also expected because acoustic songs are
much more mellow and would lack energy.
spotify %>%
ggplot(aes(x = pop)) +
geom_bar(fill = "darkseagreen") +
labs(x = "Popularity", y = "# of Songs", title = "Distribution of Popularity of Spotify Songs")
For this data, we can see that most of the popularity for these Spotify Songs are right-skewed and that these songs are mostly very popular. According to the graph, most of these songs lie within the 60-80 range.
# Grouping genres into the most popular 3 and "Other" genre
spotify$top_genre <- factor(spotify$top_genre)
spotify$top_genre <- fct_lump(spotify$top_genre, n = 5, other_level = "other")
spotify %>%
ggplot(aes(y = reorder(top_genre, top_genre, length))) +
geom_bar(fill = "darkseagreen4") +
labs(title = "Count of the Top Genres of Spotify Songs", y = "Genre")
This plot shows that in our data set of top Spotify songs, there is an overwhelmingly amount of songs that go under the dance pop genre. The second genre (besides Other) with the most popular songs is pop. It is evident that the songs in this data is mostly ones that follow into some variation of the pop genre category (ex. Electropop, Canadian Pop, etc). Pop songs have catchy rhythms and lyrics that allow people to sing along. The chorus for these songs are repetitive, have lyrics that easy to remember, and are easy to listen to.
It’s time to set up our models! Let’s finally see if we are able to predict a song’s popularity based on these components. First, we will split the data, create a recipe, and then create folds for k-fold cross validation.
To begin, we will split the data. 70% of our data will be for
training and 30% for testing. We will be stratifying against the
pop variable to make sure that the proportions are equal.
The purpose of having a training set is pretty intuitive. It will be
used to train our model. The testing data will be used to test how well
our model did by introducing data that our model has not been exposed to
before.
set.seed(100)
# Splitting the data (70% for training and 30% for testing), stratifying against the popularity
spotify_split <- spotify %>%
initial_split(prop = 0.7, strata = "pop")
# Training Data
spotify_train <- training(spotify_split)
# Testing Data
spotify_test <- testing(spotify_split)
Verifying the split:
nrow(spotify_train)/nrow(spotify)
## [1] 0.6965174
nrow(spotify_test)/nrow(spotify)
## [1] 0.3034826
This shows that there is approximately 70% of our data in the training data and approximately 30% in the testing data.
Since we will be using the same predictors and response variable in
all our models, we will create a general recipe to use. As we build this
recipe, we can think of the recipe as the song and all the different
components in the song as the ingredients. The different components can
be the guitar, drums, singer, producer, etc. Each model could be remixes
of the song since we would still be using the same recipe/song but it is
applied differently. The categorical variables we have in our data is
year but we will exclude this since it is not super
relevant to what we are analyzing. The other categorical variable is
top_genre but most of the songs are pop songs so genre
would not play a significant role in predicting popularity of a song.
Other than that, all of our predictors will be used in creating our
recipe.
spotify_recipe <- recipe(pop ~ bpm + nrgy + dnce + d_b + live + val + dur + acous + spch, data = spotify_train) %>%
# Normalizing
step_normalize(all_numeric_predictors()) %>%
# Centering
step_center(all_predictors()) %>%
# Scaling
step_scale(all_predictors())
prep(spotify_recipe) %>%
bake(new_data = spotify_train) %>%
kable() %>%
kable_styling(full_width = F) %>%
scroll_box(width = "100%", height = "200px")
| bpm | nrgy | dnce | d_b | live | val | dur | acous | spch | pop |
|---|---|---|---|---|---|---|---|---|---|
| 0.2949367 | 1.2582683 | 0.5769283 | -0.2889031 | 1.8633145 | 1.3997249 | 1.2366204 | -0.6055632 | 0.2300409 | 59 |
| 1.3646551 | 0.6324865 | -1.1079066 | 0.2861647 | -0.4516048 | -1.1304991 | 0.7837545 | -0.5573386 | 0.0960365 | 58 |
| 1.3646551 | 0.8827992 | -1.0313232 | 0.8612326 | -0.4516048 | 0.8762303 | -1.1409255 | 1.5645438 | 2.5081158 | 58 |
| -0.6036269 | -1.8706404 | 0.5003449 | -0.8639710 | -0.6009545 | -1.8284920 | 1.3498369 | -0.5091140 | -0.7079900 | 57 |
| -0.2613169 | 1.0079556 | 1.1895955 | 0.8612326 | -0.6756293 | 1.4433495 | -0.9144926 | 0.2142550 | -0.1719723 | 56 |
| -3.2137399 | -1.9957967 | -3.1756585 | -0.2889031 | -0.3022552 | -1.9593656 | 0.8969710 | 3.3970786 | -0.5739855 | 56 |
| 0.0809930 | 0.6324865 | 0.8832619 | 0.2861647 | 2.8340871 | 0.0909884 | -1.1126214 | -0.7020124 | -0.5739855 | 55 |
| -1.6305566 | 0.9453774 | -1.4908236 | 0.2861647 | 0.5938426 | 0.2654866 | 2.9348675 | -0.3644402 | 4.1161687 | 49 |
| 0.4233029 | 1.6963155 | 0.1940113 | 1.4363005 | -0.6009545 | -0.3016326 | -0.9711008 | -0.6537878 | -0.7079900 | 0 |
| -2.3579651 | -2.0583749 | -2.6395747 | -0.8639710 | -0.8249789 | -1.1304991 | 1.3781410 | 1.1305224 | -0.7079900 | 60 |
| 0.6372466 | 0.0692830 | 0.0408445 | 0.2861647 | 0.4444930 | -0.5633799 | 0.1044557 | -0.6537878 | -0.5739855 | 59 |
| -1.4594016 | 0.2570175 | -0.4186559 | 0.8612326 | -0.7503041 | 0.7889812 | -0.7163637 | -0.6055632 | 0.7660585 | 59 |
| 0.0809930 | 0.6324865 | 0.5003449 | 0.2861647 | 1.4152656 | 1.6178477 | -0.0653690 | -0.0750926 | -0.3059767 | 58 |
| -1.6305566 | 1.2582683 | -0.4952393 | 0.8612326 | 0.8178670 | 0.9198548 | 1.0384916 | -0.6055632 | -0.5739855 | 58 |
| 0.4233029 | 0.6324865 | 1.3427624 | 1.4363005 | -0.9743286 | 1.6178477 | 0.8969710 | -0.4608894 | -0.4399811 | 54 |
| -1.8445003 | -2.2461094 | -2.2566576 | -0.2889031 | -0.4516048 | -1.8721165 | 0.6139298 | 2.8183834 | -0.7079900 | 50 |
| -1.6305566 | 0.9453774 | -1.4908236 | 0.2861647 | 0.5938426 | 0.2654866 | 2.9348675 | -0.3644402 | 4.1161687 | 49 |
| 0.2949367 | -2.0583749 | -2.6395747 | -1.4390389 | -0.5262797 | -1.4358710 | 0.8403627 | 3.6864262 | -0.7079900 | 46 |
| 0.3805141 | 0.1944393 | 0.9598453 | -0.8639710 | 1.6392900 | 0.3091111 | -0.6880596 | -0.4126648 | 1.9720982 | 38 |
| 0.1665704 | -0.4313424 | 0.2705947 | -0.8639710 | -0.4516048 | 0.1782375 | -0.4899308 | -0.7020124 | 0.4980497 | 38 |
| 0.4660916 | 1.0705337 | 0.6535117 | 0.8612326 | 1.0418915 | 0.7453566 | -0.6880596 | -0.6055632 | -0.1719723 | 31 |
| 0.5088804 | 1.0705337 | -1.1079066 | 0.8612326 | 0.2951433 | -0.6506290 | 1.5196616 | 1.1787470 | 3.9821643 | 28 |
| 0.3377254 | 0.6324865 | 0.0408445 | 0.8612326 | 3.6555101 | 0.8762303 | 0.1610639 | -0.7020124 | -0.5739855 | 27 |
| 0.2521479 | 0.7576429 | -0.0357389 | 0.2861647 | -0.5262797 | 0.5708584 | 0.0761516 | -0.0268680 | -0.4399811 | 25 |
| -0.8603593 | 0.0692830 | 1.1895955 | 0.2861647 | 0.0711189 | 0.0473638 | 0.0478474 | 1.9985652 | 3.3121423 | 7 |
| 0.3805141 | 1.5711591 | 0.5769283 | 0.8612326 | 1.4899404 | 1.8795950 | -0.5465390 | -0.6537878 | 0.0960365 | 59 |
| 1.3646551 | 0.8202211 | -0.4952393 | -0.8639710 | -0.7503041 | -0.1707590 | 0.9252751 | 0.9376240 | -0.4399811 | 38 |
| -0.1757395 | 1.3208464 | -0.3420725 | 1.4363005 | 0.5191678 | 1.0071040 | 0.8686669 | -0.7020124 | -0.5739855 | 0 |
| -1.7589228 | -0.6816551 | -0.4952393 | -0.2889031 | 1.8633145 | 0.5708584 | 1.4630533 | -0.0268680 | -0.4399811 | 60 |
| 0.5088804 | -2.9970475 | -1.4908236 | -1.4390389 | -0.5262797 | -1.2613728 | -0.1219773 | 3.7346508 | -0.5739855 | 60 |
| -1.0743030 | 0.6324865 | -0.7249896 | -0.2889031 | -0.2275804 | -0.7815027 | 1.2932286 | -0.2679910 | 3.0441335 | 59 |
| 0.8511903 | -0.1810297 | -0.2654891 | -0.8639710 | 1.8633145 | -0.1271344 | 5.6237586 | -0.6537878 | 2.1061026 | 58 |
| -1.6305566 | 0.5073302 | -0.7249896 | 0.8612326 | -0.6009545 | 0.2654866 | 0.1893680 | -0.6537878 | -0.4399811 | 57 |
| 0.4233029 | 0.1318611 | 0.3471781 | 0.2861647 | 1.1165663 | 0.5708584 | 0.5290174 | -0.6537878 | 1.1680718 | 57 |
| -0.6036269 | 1.4460028 | -0.7249896 | 0.8612326 | 0.3698181 | 0.1782375 | -0.5465390 | -0.6537878 | 0.6320541 | 56 |
| 0.7656128 | -1.6203277 | -2.7927415 | -1.4390389 | -0.5262797 | -1.7848674 | 0.3025845 | 2.3361374 | -0.7079900 | 55 |
| -0.7747818 | 0.4447520 | -0.1123223 | -0.8639710 | -1.0490034 | -1.0432500 | -0.6314514 | -0.7020124 | -0.5739855 | 54 |
| -0.4752606 | -0.3061861 | -0.4186559 | 0.2861647 | 3.4314856 | -0.8251272 | 3.6141663 | -0.5091140 | 1.0340673 | 54 |
| 0.4233029 | 1.0705337 | -0.1889057 | 0.2861647 | 1.2659159 | 0.6144830 | -0.1219773 | -0.6537878 | 0.2300409 | 52 |
| 1.2790777 | 1.6963155 | 0.2705947 | 2.0113684 | 0.7431922 | 1.6614722 | -0.7729720 | -0.3644402 | 0.0960365 | 52 |
| -0.4752606 | 0.3195957 | 0.2705947 | 0.2861647 | 0.9672167 | -0.3016326 | -0.7163637 | 0.6965010 | -0.5739855 | 51 |
| 3.5468808 | 1.5085810 | -2.1800742 | 1.4363005 | 1.4152656 | -0.0398853 | -0.4050184 | -0.2679910 | 5.3222084 | 50 |
| 0.4233029 | 0.2570175 | 0.5769283 | 0.8612326 | -0.3769300 | -0.0835099 | 0.1044557 | -0.5091140 | -0.5739855 | 46 |
| 0.4233029 | -0.8693896 | 0.3471781 | 0.2861647 | -0.5262797 | 0.9198548 | 0.8969710 | -0.6055632 | -0.4399811 | 45 |
| 0.4233029 | 0.5073302 | -0.3420725 | 0.2861647 | 1.3405908 | 0.3963602 | 0.9535792 | -0.5091140 | -0.5739855 | 45 |
| -0.4752606 | -0.0558734 | 0.9598453 | -0.2889031 | -0.8996538 | 1.1816022 | 0.0478474 | -0.6537878 | -0.7079900 | 43 |
| 0.5088804 | 0.1944393 | -0.8781564 | -0.2889031 | 0.4444930 | -0.3016326 | -1.1126214 | -0.7020124 | -0.5739855 | 42 |
| 0.5944578 | -2.3086876 | -1.0313232 | -3.1642426 | -0.4516048 | -1.9593656 | 0.6988421 | 0.4071534 | -0.7079900 | 36 |
| -0.6036269 | 0.4447520 | 0.0408445 | 0.2861647 | -0.4516048 | -0.3016326 | 1.3781410 | -0.0750926 | -0.3059767 | 28 |
| 0.0809930 | 0.5699084 | 2.1851798 | -0.2889031 | -0.6009545 | 1.6614722 | -0.6597555 | 0.6000518 | -0.0379679 | 59 |
| 2.2204299 | 0.5073302 | -1.3376568 | -2.0141068 | 0.9672167 | -1.3486219 | -0.2634979 | 0.1178058 | 2.3741114 | 56 |
| 0.2949367 | 1.4460028 | -0.3420725 | 0.8612326 | -0.6009545 | 0.1346129 | -0.0936732 | -0.7020124 | -0.1719723 | 56 |
| -0.0901620 | -3.5602510 | -1.7971572 | -2.5891747 | -0.6756293 | -1.2613728 | -0.9994049 | 3.9757738 | -0.5739855 | 56 |
| 0.9367677 | -0.5564988 | -0.4186559 | -0.8639710 | -0.0035559 | -0.5197554 | 2.7650427 | -0.6537878 | -0.4399811 | 52 |
| 0.4233029 | 0.6950647 | 1.1895955 | 0.2861647 | 1.2659159 | 0.3963602 | -0.7729720 | 0.1660304 | -0.5739855 | 50 |
| 0.0809930 | -2.2461094 | -0.8781564 | -2.0141068 | -0.3022552 | -1.4358710 | 0.3874969 | 0.6482764 | -0.7079900 | 44 |
| 1.3218664 | -0.8068114 | -0.1123223 | -0.8639710 | 1.4899404 | 0.1782375 | -0.0370649 | 0.6482764 | -0.1719723 | 38 |
| 0.5088804 | -1.3074368 | 1.0364287 | -0.8639710 | -0.5262797 | -0.5633799 | 0.2742804 | 0.0213566 | -0.5739855 | 37 |
| -1.1170917 | 0.6950647 | -0.6484062 | 1.4363005 | 0.0711189 | 0.2218620 | -0.2918020 | -0.6055632 | -0.4399811 | 37 |
| -1.3738241 | 0.1318611 | -0.4186559 | -0.2889031 | 1.4899404 | -0.2580081 | 1.1517080 | 0.6000518 | -0.3059767 | 0 |
| 0.2093592 | 0.6950647 | -0.0357389 | 0.2861647 | -0.1529056 | 0.5272339 | -0.1785855 | -0.6055632 | -0.3059767 | 60 |
| -1.6305566 | -1.4325932 | 0.3471781 | -1.4390389 | -0.8996538 | -1.0432500 | 0.1327598 | 1.0822978 | -0.0379679 | 56 |
| 1.6213876 | -0.6816551 | -1.3376568 | -0.2889031 | -0.6756293 | -2.0902393 | 1.8876151 | 0.8893994 | -0.4399811 | 55 |
| -0.7747818 | 0.6950647 | 0.0408445 | 0.2861647 | -0.4516048 | -0.5197554 | 0.1327598 | -0.4126648 | -0.0379679 | 55 |
| -1.2454579 | -0.7442333 | -1.2610734 | -0.2889031 | -0.8996538 | -1.0868746 | 2.3687851 | 0.2624796 | -0.4399811 | 52 |
| 0.2093592 | 0.5073302 | 0.1940113 | 0.2861647 | 0.5191678 | -0.5633799 | -0.1785855 | -0.5091140 | 0.0960365 | 52 |
| -0.7747818 | -0.9945460 | 0.3471781 | 0.2861647 | -0.6009545 | -1.1304991 | -0.6314514 | -0.6537878 | -0.7079900 | 52 |
| -1.7589228 | 0.3195957 | -1.3376568 | -0.2889031 | -0.3022552 | 0.0037392 | 0.0478474 | -0.7020124 | -0.5739855 | 51 |
| -1.0743030 | -0.2436079 | -0.0357389 | -0.2889031 | -0.7503041 | -1.9157411 | 1.6894863 | 1.3716454 | -0.5739855 | 49 |
| -0.0473732 | -1.7454840 | -2.9459083 | -3.1642426 | -0.5262797 | -1.8721165 | -0.0087608 | 2.9148326 | -0.7079900 | 47 |
| 0.4233029 | 0.4447520 | -0.5718228 | 0.8612326 | 1.1165663 | -1.0432500 | 0.5007133 | -0.6055632 | -0.7079900 | 46 |
| 0.4233029 | 0.4447520 | 1.1895955 | -0.2889031 | -1.0490034 | 1.3561004 | -0.8295802 | -0.6537878 | 1.3020762 | 42 |
| 0.3377254 | 0.8202211 | -0.1889057 | 0.2861647 | -0.7503041 | -0.5633799 | -0.0370649 | -0.6055632 | -0.0379679 | 41 |
| 0.0809930 | 0.5699084 | 1.8022628 | 0.8612326 | 0.1457937 | 0.0909884 | -0.6597555 | -0.3162156 | -0.3059767 | 39 |
| -0.2613169 | -2.1209531 | -2.7927415 | -0.8639710 | -0.5262797 | -1.9157411 | 1.3498369 | 1.4680946 | -0.5739855 | 36 |
| -0.2613169 | 1.1956901 | 0.5769283 | 0.8612326 | -0.8996538 | 1.5742231 | 0.3591927 | -0.6537878 | 0.6320541 | 34 |
| -0.3896832 | -1.1197023 | 1.9554296 | 0.2861647 | -0.7503041 | 0.5708584 | 0.8403627 | -0.7020124 | -0.4399811 | 18 |
| 0.3377254 | 0.4447520 | 0.5003449 | 0.2861647 | 0.8925419 | 0.5272339 | -0.6314514 | 0.7447256 | -0.3059767 | 58 |
| -0.1757395 | 0.3821738 | 0.2705947 | 0.2861647 | -0.1529056 | 0.8762303 | -0.0370649 | -0.1715418 | 0.2300409 | 57 |
| -1.0315142 | -0.8693896 | -2.1034908 | -0.2889031 | -0.3769300 | -1.7412428 | 5.0293722 | 2.7701588 | -0.7079900 | 57 |
| -1.5449791 | -1.0571241 | -0.1889057 | -0.2889031 | -0.3769300 | -1.3922464 | -1.0843173 | 2.5772604 | -0.0379679 | 54 |
| 0.2093592 | -0.0558734 | 0.4237615 | -0.8639710 | 0.2951433 | 0.2654866 | 0.1610639 | 0.0695812 | 0.2300409 | 53 |
| 0.3377254 | 0.5073302 | -0.5718228 | 1.4363005 | -0.6009545 | -0.1271344 | -0.0087608 | 1.2269716 | -0.5739855 | 51 |
| -0.0045845 | 0.0067048 | 0.7300951 | 0.2861647 | -0.3769300 | 1.2252267 | -0.2351937 | -0.5091140 | -0.4399811 | 50 |
| 0.5088804 | 1.5085810 | -0.1123223 | 0.2861647 | -0.6009545 | 0.9198548 | -0.8012761 | -0.7020124 | -0.5739855 | 48 |
| -0.3468944 | -4.1860328 | -0.9547398 | -5.4645141 | -0.6756293 | -1.1304991 | 2.3970892 | 4.0722230 | -0.5739855 | 47 |
| -1.2026692 | 0.2570175 | 1.4193458 | 0.8612326 | 0.1457937 | -0.8687518 | 1.0667957 | 0.3589288 | 0.4980497 | 47 |
| -0.1757395 | 1.0079556 | 0.0408445 | 0.2861647 | 1.4152656 | -0.3452572 | 0.2459763 | -0.6537878 | -0.3059767 | 46 |
| 0.2521479 | 0.3195957 | -0.5718228 | -0.2889031 | -1.0490034 | -0.5197554 | -0.6031473 | -0.7020124 | -0.3059767 | 36 |
| 0.1237817 | -0.0558734 | 0.8066785 | -0.2889031 | -0.5262797 | -0.3016326 | -0.6314514 | -0.6537878 | -0.1719723 | 34 |
| 0.2093592 | 0.0067048 | 1.4193458 | -0.8639710 | 0.9672167 | 1.3561004 | 0.6988421 | -0.4608894 | 1.3020762 | 31 |
| 2.9050497 | 0.6324865 | 0.4237615 | 1.4363005 | 1.4152656 | 0.7453566 | -0.0087608 | -0.2679910 | -0.1719723 | 29 |
| 0.2521479 | -1.1197023 | 0.7300951 | -0.2889031 | 1.2659159 | -1.6976183 | -1.0277090 | -0.7020124 | -0.5739855 | 59 |
| 2.0920637 | -1.3074368 | -2.1034908 | -2.0141068 | 0.1457937 | -1.3486219 | -1.2824461 | -0.4126648 | 0.4980497 | 59 |
| -0.3468944 | 1.1956901 | 0.7300951 | 2.0113684 | -0.7503041 | 0.5708584 | -1.1692296 | -0.3644402 | -0.4399811 | 55 |
| -0.7747818 | -0.2436079 | -1.0313232 | -2.0141068 | 1.3405908 | -1.0868746 | 1.0950998 | 2.0950144 | -0.3059767 | 54 |
| -0.1329507 | 0.7576429 | 0.4237615 | -0.2889031 | 1.3405908 | 0.4836093 | 0.5573216 | 0.6482764 | 2.5081158 | 54 |
| -0.5608381 | -0.7442333 | 0.9598453 | -2.5891747 | -0.5262797 | -0.8251272 | -0.0087608 | -0.2197664 | -0.4399811 | 54 |
| -0.2185282 | -1.1197023 | 0.2705947 | -0.2889031 | -1.0490034 | 1.3997249 | -0.6597555 | -0.0268680 | 2.6421202 | 53 |
| 1.7069650 | 0.6324865 | -0.5718228 | 0.2861647 | -1.0490034 | 1.1379776 | -0.0653690 | -0.6055632 | -0.4399811 | 51 |
| -0.9887255 | 0.8202211 | -0.3420725 | -0.8639710 | -0.5262797 | -0.2143835 | 0.4158010 | -0.6055632 | -0.1719723 | 49 |
| -1.3310354 | -1.5577495 | -2.8693249 | 0.8612326 | -0.3022552 | -1.3486219 | 1.7177904 | 3.3970786 | -0.7079900 | 44 |
| 0.0809930 | 0.3821738 | 0.8832619 | -0.2889031 | -0.7503041 | 1.2688513 | -0.6597555 | -0.3162156 | -0.4399811 | 43 |
| 1.2790777 | 1.0705337 | -1.6439904 | 0.2861647 | 0.2204685 | -1.1741237 | 1.6045739 | -0.7020124 | -0.5739855 | 41 |
| 0.5088804 | -0.1184515 | -0.1123223 | 0.8612326 | -0.4516048 | -1.1741237 | 0.0478474 | 0.0695812 | 0.0960365 | 57 |
| -1.7589228 | -1.4325932 | -0.4952393 | -0.2889031 | -0.6009545 | -0.4325063 | -0.3484102 | -0.6055632 | -0.7079900 | 57 |
| 0.0809930 | -0.9945460 | 1.1130121 | -0.8639710 | -0.3769300 | -1.2613728 | 0.0195433 | -0.6537878 | -0.4399811 | 57 |
| 1.7925425 | 0.8202211 | -0.4952393 | 0.2861647 | -0.6009545 | -0.0835099 | -0.9994049 | -0.0750926 | 1.8380938 | 52 |
| -0.6892043 | 0.3821738 | -0.4952393 | 0.2861647 | -0.6756293 | 1.0507285 | -1.2258379 | -0.4608894 | -0.0379679 | 44 |
| -0.8175705 | 0.5699084 | -0.1123223 | -0.8639710 | 1.3405908 | -0.6506290 | 1.3781410 | -0.7020124 | -0.4399811 | 69 |
| 0.5088804 | -0.4939206 | 1.1895955 | -0.8639710 | 1.0418915 | -0.5197554 | 1.6611822 | -0.7020124 | -0.1719723 | 68 |
| 0.2949367 | -0.1810297 | 0.6535117 | 0.2861647 | -0.6756293 | 0.9634794 | -0.2918020 | -0.7020124 | -0.7079900 | 66 |
| 1.1935002 | 1.5085810 | -0.8781564 | 0.8612326 | 0.7431922 | 0.5708584 | 0.3591927 | 0.5518272 | -0.4399811 | 65 |
| 1.2790777 | 0.7576429 | -1.2610734 | -0.2889031 | 1.4899404 | 0.9634794 | -0.3767143 | -0.7020124 | -0.5739855 | 65 |
| 0.0809930 | 0.5699084 | 0.5769283 | 0.2861647 | -0.3769300 | 0.3091111 | 0.0761516 | -0.6055632 | -0.5739855 | 63 |
| 0.2949367 | 0.2570175 | 1.1130121 | 0.8612326 | -0.0035559 | -0.5197554 | 0.2742804 | -0.7020124 | -0.4399811 | 62 |
| 0.7228241 | 0.0067048 | 0.4237615 | 2.0113684 | -0.3022552 | 1.1816022 | -0.3484102 | -0.4608894 | -0.4399811 | 62 |
| 0.8511903 | 1.1331119 | 0.2705947 | 0.8612326 | 1.3405908 | 1.3561004 | -0.9427967 | -0.6537878 | -0.0379679 | 62 |
| 0.0809930 | 0.8202211 | 0.8066785 | 1.4363005 | 1.7886397 | 0.0037392 | -1.5088790 | -0.3162156 | 0.4980497 | 62 |
| -1.1170917 | -1.1822805 | -0.3420725 | -0.8639710 | 0.9672167 | -0.4761308 | 0.9252751 | 0.1660304 | 3.8481599 | 62 |
| -0.3041057 | 0.6950647 | -1.1079066 | 0.2861647 | -0.3769300 | 0.1346129 | 0.6705380 | 0.0695812 | 0.3640453 | 62 |
| 0.5088804 | -0.4939206 | 1.2661790 | 0.2861647 | -0.3769300 | 1.2252267 | -0.8295802 | -0.3162156 | -0.4399811 | 61 |
| 1.9636975 | -1.8706404 | -2.7161581 | 0.2861647 | -0.3769300 | -1.9593656 | 1.0950998 | 3.3006294 | -0.7079900 | 61 |
| 0.5088804 | 0.4447520 | 0.6535117 | 0.2861647 | -0.8249789 | 0.2218620 | 1.6894863 | -0.1715418 | -0.4399811 | 69 |
| 0.5088804 | 1.0705337 | 0.5003449 | 1.4363005 | -0.6756293 | 0.7889812 | 0.4724092 | -0.7020124 | -0.5739855 | 69 |
| 0.4233029 | 0.3821738 | -0.4952393 | -0.2889031 | -0.5262797 | -0.6942536 | 2.7084345 | -0.7020124 | -0.5739855 | 67 |
| 0.5944578 | 0.0067048 | 0.3471781 | -0.2889031 | 0.1457937 | -0.3016326 | 0.3591927 | -0.6055632 | -0.1719723 | 65 |
| 0.5088804 | -0.9945460 | 0.4237615 | -0.8639710 | 1.1165663 | 1.1816022 | -0.4333226 | -0.7020124 | -0.5739855 | 64 |
| 0.5516691 | 1.3834246 | 0.1174279 | 0.8612326 | 0.6685174 | 0.0473638 | 0.6705380 | -0.7020124 | -0.1719723 | 63 |
| 0.3377254 | 1.1331119 | 0.0408445 | 0.8612326 | -0.4516048 | -0.8687518 | 0.1893680 | -0.5573386 | -0.4399811 | 63 |
| 0.3805141 | -0.9945460 | 0.8832619 | 0.2861647 | -1.0490034 | 1.1379776 | 0.1893680 | -0.6537878 | -0.7079900 | 69 |
| 0.4233029 | 0.8827992 | 0.5769283 | -0.2889031 | 0.2951433 | -0.3452572 | 0.5007133 | -0.7020124 | 0.0960365 | 68 |
| 1.7925425 | -0.1810297 | -1.0313232 | 0.2861647 | 1.5646152 | 0.2218620 | -0.1219773 | -0.3644402 | -0.5739855 | 68 |
| 0.6800353 | 0.5699084 | 0.3471781 | 0.8612326 | -0.1529056 | 1.5742231 | 0.0478474 | -0.7020124 | -0.5739855 | 67 |
| 0.0809930 | 0.5699084 | 0.2705947 | -0.2889031 | -0.3769300 | 1.1379776 | -0.4050184 | -0.5573386 | -0.5739855 | 65 |
| 2.8194723 | -0.4313424 | -2.0269074 | -0.2889031 | -0.7503041 | 0.9634794 | 0.7271463 | -0.6055632 | -0.0379679 | 63 |
| -0.9887255 | -1.1197023 | 1.0364287 | -0.8639710 | -0.9743286 | 0.6144830 | 0.3025845 | -0.4126648 | 2.7761247 | 69 |
| 0.3377254 | 0.2570175 | -0.9547398 | -0.2889031 | -0.3022552 | -1.5667446 | -0.2918020 | 0.5518272 | -0.5739855 | 69 |
| -1.5449791 | 1.0705337 | 0.0408445 | 1.4363005 | -0.7503041 | 1.4869740 | 0.0478474 | 0.6965010 | 0.3640453 | 68 |
| -0.6892043 | -0.6816551 | 1.1895955 | 0.8612326 | -0.1529056 | -0.6942536 | 2.8499551 | -0.2197664 | 1.8380938 | 68 |
| 0.3377254 | 1.2582683 | 0.5003449 | 1.4363005 | -0.2275804 | 1.3124758 | -1.9334408 | -0.6537878 | -0.7079900 | 67 |
| 0.5088804 | 1.0079556 | 0.5769283 | 1.4363005 | 3.5808353 | 0.7017321 | -0.4616267 | -0.5091140 | -0.5739855 | 67 |
| 0.0809930 | -0.2436079 | 0.1174279 | 0.2861647 | 0.5938426 | -0.6506290 | -0.0087608 | -0.7020124 | -0.7079900 | 66 |
| 0.0809930 | 0.4447520 | 0.2705947 | -0.2889031 | -0.4516048 | -0.5197554 | 0.1327598 | -0.0750926 | -0.5739855 | 65 |
| 0.1237817 | 0.1944393 | 1.5725126 | 0.2861647 | 2.7594123 | 1.0943531 | -0.0370649 | -0.7020124 | 0.0960365 | 63 |
| 0.3377254 | 0.6950647 | -0.6484062 | 1.4363005 | -0.6009545 | -0.3888817 | -0.6597555 | -0.6537878 | -0.3059767 | 62 |
| -0.5608381 | -1.1822805 | -0.6484062 | -0.8639710 | 1.1912411 | 0.2654866 | 1.0667957 | -0.0268680 | 0.2300409 | 61 |
| 0.9367677 | -1.0571241 | 0.5769283 | -0.8639710 | -0.6009545 | 0.5272339 | 0.5290174 | 1.4680946 | -0.5739855 | 68 |
| 1.1079227 | -1.3700150 | -1.5674070 | -0.2889031 | -0.5262797 | -1.1741237 | 0.7271463 | -0.0750926 | -0.5739855 | 68 |
| 0.3377254 | 0.2570175 | -0.6484062 | 0.2861647 | 0.2204685 | -0.0835099 | 0.1044557 | 1.5163192 | -0.7079900 | 67 |
| -1.5449791 | -0.5564988 | 0.4237615 | -0.8639710 | -0.1529056 | 0.8762303 | -0.9144926 | -0.3162156 | 2.2401070 | 67 |
| 0.0809930 | 1.3834246 | 0.5769283 | 0.8612326 | -0.7503041 | 0.9634794 | -0.0087608 | 0.1178058 | -0.3059767 | 66 |
| -0.9459368 | -0.4313424 | 0.3471781 | -0.2889031 | -0.2275804 | 1.2252267 | -0.6031473 | -0.6055632 | -0.4399811 | 66 |
| 0.3377254 | 1.3208464 | 0.5003449 | 0.8612326 | 0.6685174 | 2.0104687 | 0.1893680 | -0.7020124 | -0.5739855 | 65 |
| 2.9050497 | -1.6829059 | -1.7971572 | -3.7393104 | 0.9672167 | -1.5231201 | 2.6518263 | -0.5573386 | 2.7761247 | 65 |
| 0.3377254 | -0.2436079 | 0.5769283 | 0.2861647 | -0.5262797 | 1.3997249 | -0.2918020 | -0.2679910 | -0.1719723 | 65 |
| -0.3468944 | 1.3208464 | -0.1889057 | 1.4363005 | -0.4516048 | 0.2218620 | 0.2176721 | -0.6537878 | -0.5739855 | 65 |
| 0.4233029 | 0.1944393 | -0.3420725 | 1.4363005 | -0.6756293 | -0.2580081 | -0.2351937 | -0.1715418 | -0.5739855 | 64 |
| -1.5021904 | 0.0067048 | -1.3376568 | 0.2861647 | -0.1529056 | -1.3049973 | 0.7554504 | -0.3162156 | -0.7079900 | 62 |
| 0.8511903 | 0.6324865 | 0.3471781 | 0.8612326 | -0.3022552 | 1.3124758 | -0.5182349 | -0.1233172 | -0.1719723 | 62 |
| 3.1617822 | 0.3821738 | -2.2566576 | 0.8612326 | -0.4516048 | -2.0902393 | -0.9711008 | -0.4126648 | -0.4399811 | 61 |
| 0.8511903 | -3.4976729 | -1.4908236 | -2.0141068 | -0.6756293 | -1.8721165 | 0.1044557 | 3.4935278 | -0.7079900 | 61 |
| -1.1598805 | 1.1331119 | -0.5718228 | 0.2861647 | 1.1165663 | -0.9123764 | 0.2176721 | -0.2197664 | -0.0379679 | 69 |
| 1.3218664 | 0.5699084 | -1.1844900 | 0.8612326 | -0.8996538 | -0.1707590 | -0.7446679 | -0.6537878 | -0.4399811 | 69 |
| -0.1329507 | 0.8827992 | 0.3471781 | 0.2861647 | -0.6009545 | 0.5272339 | -0.0370649 | -0.6055632 | -0.7079900 | 69 |
| -1.5449791 | -0.0558734 | -0.1123223 | 0.2861647 | -0.6009545 | 0.5272339 | -0.2351937 | 1.9985652 | 2.1061026 | 69 |
| 0.8939790 | 0.8827992 | -2.4864079 | -0.2889031 | 2.2366886 | -1.9157411 | -0.3484102 | -0.6537878 | -0.0379679 | 69 |
| 0.5088804 | 0.8827992 | -0.4952393 | 1.4363005 | -0.3022552 | -0.6506290 | 0.4158010 | -0.6537878 | -0.3059767 | 68 |
| 0.0809930 | 0.6950647 | -2.0269074 | 0.8612326 | -0.7503041 | -0.8251272 | -0.7446679 | -0.3162156 | 0.2300409 | 68 |
| 0.0809930 | 0.8827992 | -0.4186559 | 0.2861647 | 0.6685174 | -0.7815027 | 0.2742804 | -0.3644402 | -0.5739855 | 67 |
| 0.2093592 | 1.4460028 | 0.3471781 | 0.2861647 | -0.9743286 | 1.1816022 | 0.2742804 | -0.2679910 | -0.3059767 | 67 |
| 0.2521479 | 1.4460028 | 0.2705947 | 0.8612326 | -0.4516048 | 0.5708584 | 0.1327598 | 0.1660304 | -0.3059767 | 67 |
| 0.3805141 | 0.6324865 | -0.8781564 | -0.2889031 | -0.6009545 | -1.3486219 | 0.5573216 | -0.3162156 | -0.3059767 | 67 |
| 0.1665704 | 0.6324865 | -0.1889057 | 0.2861647 | 0.2951433 | 0.3527357 | -0.5748432 | -0.6537878 | -0.5739855 | 66 |
| -0.6464156 | 1.0079556 | 1.2661790 | 0.2861647 | -0.9743286 | 0.9634794 | -1.5371832 | -0.6537878 | 0.3640453 | 66 |
| 0.0809930 | 0.5073302 | 0.8066785 | -0.8639710 | -0.6756293 | 1.5742231 | 0.2742804 | -0.4126648 | -0.7079900 | 66 |
| 0.6800353 | 1.0705337 | 1.2661790 | 0.8612326 | -0.5262797 | 1.9232196 | -1.0277090 | -0.4608894 | -0.4399811 | 65 |
| 1.5785988 | 0.0067048 | 0.4237615 | -0.8639710 | -0.6756293 | 0.9634794 | -0.3484102 | 0.5518272 | -0.4399811 | 65 |
| -0.5180494 | -0.2436079 | 1.1130121 | 0.8612326 | 0.7431922 | 1.4433495 | -0.7729720 | 0.3107042 | -0.5739855 | 65 |
| 0.4660916 | 1.5085810 | 0.5003449 | 0.8612326 | 0.5938426 | -0.5197554 | -0.4333226 | -0.7020124 | -0.4399811 | 64 |
| 0.2949367 | -2.1835312 | 0.9598453 | -3.1642426 | -0.6009545 | 1.0507285 | -0.6031473 | 3.1077310 | -0.5739855 | 64 |
| -0.1757395 | -2.8093130 | -2.0269074 | -2.0141068 | -0.0782308 | 0.0037392 | -2.5844355 | 3.5417524 | -0.7079900 | 63 |
| 0.2093592 | 0.2570175 | 0.1940113 | -0.8639710 | 0.0711189 | -0.1707590 | 0.4724092 | -0.0268680 | -0.5739855 | 62 |
| 1.3646551 | 0.2570175 | 0.5003449 | 0.8612326 | 1.4899404 | 0.9198548 | -0.7446679 | 0.6000518 | 0.0960365 | 61 |
| 0.1665704 | -0.0558734 | 0.5003449 | -1.4390389 | -0.8249789 | 0.3527357 | -1.8768326 | -0.6055632 | -0.7079900 | 61 |
| -0.6892043 | -0.1184515 | 0.1940113 | -0.8639710 | -0.7503041 | -1.3922464 | -0.7446679 | 0.7929502 | 0.3640453 | 69 |
| 0.8939790 | 0.8827992 | -2.4864079 | -0.2889031 | 2.2366886 | -1.9157411 | -0.3484102 | -0.6537878 | -0.0379679 | 69 |
| 0.2949367 | -0.0558734 | -0.4186559 | 0.8612326 | -0.6756293 | -0.3888817 | -0.6031473 | -0.2197664 | -0.4399811 | 66 |
| -0.7319931 | -0.4313424 | -0.7249896 | -0.2889031 | -0.5262797 | -0.3452572 | -0.6314514 | -0.4608894 | -0.4399811 | 66 |
| -0.7747818 | -0.1184515 | -0.7249896 | -0.2889031 | 0.6685174 | -0.7378781 | -1.5654873 | -0.2679910 | -0.5739855 | 66 |
| -1.3310354 | 0.1944393 | -0.1123223 | 0.2861647 | 0.7431922 | -1.1304991 | -0.8861884 | -0.7020124 | -0.5739855 | 65 |
| -1.2026692 | 0.5699084 | -0.4186559 | 0.2861647 | -0.7503041 | 0.9198548 | -0.0087608 | -0.4126648 | 1.8380938 | 64 |
| -0.8603593 | 1.1331119 | -0.2654891 | 0.2861647 | -0.6756293 | -0.8687518 | -0.9427967 | -0.4608894 | -0.4399811 | 64 |
| 0.0382042 | 0.0692830 | -0.6484062 | -0.2889031 | -0.9743286 | -1.1741237 | 0.7271463 | -0.6537878 | -0.4399811 | 63 |
| -0.3468944 | 1.0705337 | -0.3420725 | 0.8612326 | -0.1529056 | -0.8687518 | 0.0761516 | -0.6055632 | 0.4980497 | 63 |
| 1.9636975 | -1.1822805 | 0.3471781 | -1.4390389 | -0.0782308 | 0.1782375 | -0.0653690 | -0.5091140 | 0.2300409 | 61 |
| -0.5180494 | -1.3074368 | 1.7256794 | 0.2861647 | -0.7503041 | 0.1346129 | -0.6597555 | 0.2624796 | -0.4399811 | 69 |
| 0.1665704 | 0.6324865 | 0.0408445 | -0.2889031 | -0.3769300 | 0.2654866 | -1.0277090 | -0.1233172 | -0.3059767 | 69 |
| -0.5180494 | 0.5073302 | 1.1895955 | -0.2889031 | 0.2951433 | 0.0037392 | 0.0761516 | -0.0750926 | -0.3059767 | 68 |
| 0.7228241 | -0.8693896 | 0.8832619 | -2.0141068 | 0.1457937 | -1.6539937 | 1.0101874 | -0.6055632 | 0.6320541 | 67 |
| -0.4324719 | 1.3834246 | 0.5769283 | 0.8612326 | -0.9743286 | 0.0909884 | -0.5748432 | -0.5573386 | -0.4399811 | 66 |
| 0.0809930 | 0.1944393 | 0.9598453 | -0.2889031 | -0.6756293 | 1.4869740 | -0.6597555 | -0.4608894 | -0.5739855 | 66 |
| -1.3738241 | -1.8080622 | 0.7300951 | -0.8639710 | -0.8996538 | 1.4433495 | -1.0560132 | -0.6537878 | -0.4399811 | 64 |
| 0.1665704 | 1.0079556 | -0.7249896 | 0.2861647 | -0.3022552 | -0.8251272 | 0.6705380 | -0.7020124 | -0.3059767 | 62 |
| -0.5608381 | 0.0692830 | 0.8832619 | -0.2889031 | 0.3698181 | -0.2143835 | -0.4333226 | -0.6055632 | -0.0379679 | 68 |
| -0.9887255 | 0.1318611 | 2.1085964 | -0.8639710 | -0.4516048 | -1.3922464 | -0.1502814 | -0.7020124 | 0.0960365 | 68 |
| -0.2185282 | 0.5699084 | -0.8015730 | 0.2861647 | 1.7886397 | -1.4794955 | 0.5007133 | -0.6055632 | -0.7079900 | 67 |
| 0.3377254 | -0.6816551 | 0.1174279 | 0.2861647 | -0.5262797 | -0.3016326 | -0.9711008 | -0.2197664 | -0.3059767 | 67 |
| -0.1757395 | -0.7442333 | -1.9503240 | -0.2889031 | -0.6756293 | -1.4358710 | 1.3215327 | -0.5573386 | -0.3059767 | 66 |
| -1.5449791 | -1.6829059 | -1.2610734 | -2.5891747 | 3.4314856 | 0.4399848 | 2.2272645 | 3.2041802 | 2.1061026 | 65 |
| -1.0315142 | -0.1184515 | 0.6535117 | 0.8612326 | -0.6009545 | -0.4325063 | -0.1219773 | -0.6055632 | -0.4399811 | 64 |
| -0.7747818 | 1.0079556 | 0.8832619 | 0.8612326 | -0.8249789 | 1.6614722 | -1.5937914 | -0.5573386 | -0.4399811 | 64 |
| -0.7747818 | -0.4313424 | 0.8066785 | 0.2861647 | -0.5262797 | -0.3016326 | -1.1126214 | 0.5036026 | 0.0960365 | 64 |
| 1.3646551 | -1.6829059 | 0.4237615 | -2.0141068 | -0.3769300 | -1.2613728 | -0.3484102 | -0.4126648 | -0.3059767 | 67 |
| -1.0743030 | -2.1209531 | -1.2610734 | -1.4390389 | -0.4516048 | -1.6539937 | -0.2634979 | 2.8666080 | -0.7079900 | 76 |
| 0.3377254 | 0.0692830 | 1.1130121 | 0.8612326 | -0.8249789 | 0.3963602 | 0.2742804 | -0.0750926 | -0.5739855 | 73 |
| 1.1507114 | 0.7576429 | -0.1889057 | 0.2861647 | -0.6009545 | -0.1707590 | 0.1327598 | 0.8893994 | -0.5739855 | 73 |
| 0.5088804 | 0.6950647 | 0.9598453 | 0.2861647 | 3.8795346 | 0.4836093 | -0.3484102 | 0.1660304 | -0.4399811 | 73 |
| 0.2949367 | 0.8202211 | -1.5674070 | -1.4390389 | -0.4516048 | 1.1379776 | -0.2351937 | -0.6537878 | 4.9201952 | 72 |
| 0.5088804 | 0.2570175 | 1.0364287 | -0.2889031 | 1.3405908 | -0.8251272 | 1.2366204 | 0.2624796 | -0.7079900 | 72 |
| 0.3805141 | 1.1956901 | 0.6535117 | 0.8612326 | 1.4152656 | 1.0507285 | 0.3025845 | -0.7020124 | 0.7660585 | 76 |
| 0.4233029 | 0.3195957 | 0.5769283 | 0.8612326 | 1.0418915 | 0.4399848 | -0.6880596 | -0.6537878 | -0.4399811 | 76 |
| -1.0315142 | 0.2570175 | 0.0408445 | 0.2861647 | 3.1327864 | 0.5708584 | 1.1800121 | -0.3162156 | 0.0960365 | 76 |
| -0.7747818 | 0.4447520 | 0.3471781 | 0.2861647 | -0.1529056 | 1.2688513 | -0.7163637 | -0.3162156 | 0.4980497 | 75 |
| 1.3646551 | 1.3834246 | -0.4952393 | 0.8612326 | -0.3022552 | 1.5742231 | -0.1219773 | -0.7020124 | -0.4399811 | 74 |
| 0.2521479 | 0.7576429 | -0.4186559 | 0.2861647 | 1.1165663 | -0.1271344 | 0.9818833 | -0.7020124 | 1.0340673 | 73 |
| 0.2949367 | -0.1184515 | -0.1123223 | -0.2889031 | 3.6555101 | -1.8284920 | 4.3500733 | -0.7020124 | 1.5700850 | 73 |
| -0.8175705 | -0.3687642 | -0.7249896 | 0.8612326 | -0.6756293 | -0.9560009 | 0.8120586 | -0.0268680 | -0.7079900 | 73 |
| 0.5088804 | 0.1944393 | 0.8066785 | 0.8612326 | 0.6685174 | -0.7378781 | 1.0384916 | -0.6055632 | 1.0340673 | 72 |
| 0.0809930 | 0.6950647 | 0.7300951 | 0.2861647 | -0.4516048 | 0.5708584 | -0.5748432 | -0.6537878 | -0.5739855 | 72 |
| -0.9887255 | -0.1184515 | -0.1123223 | -1.4390389 | -0.3769300 | 0.5708584 | 0.7837545 | -0.3162156 | -0.4399811 | 72 |
| -1.0315142 | -0.4939206 | 0.1174279 | -2.0141068 | -0.5262797 | -1.1304991 | -0.1785855 | 0.2142550 | 1.3020762 | 71 |
| 0.0809930 | -0.8068114 | 1.0364287 | -0.8639710 | -0.5262797 | 0.6144830 | -0.9144926 | -0.6537878 | -0.5739855 | 76 |
| -0.0473732 | -0.1810297 | 1.6490960 | 0.8612326 | -0.8249789 | 1.7487213 | -1.0560132 | -0.3162156 | -0.4399811 | 76 |
| -0.4324719 | -0.1810297 | -0.2654891 | 0.8612326 | -0.3769300 | -0.2143835 | 0.4724092 | -0.7020124 | -0.7079900 | 76 |
| 0.2949367 | 0.0692830 | 0.8066785 | 2.0113684 | 0.5191678 | 1.0071040 | -0.4050184 | -0.0268680 | -0.0379679 | 73 |
| 0.3377254 | 1.0079556 | -0.6484062 | 0.8612326 | 1.4152656 | 0.7017321 | -0.5465390 | -0.6055632 | 0.7660585 | 73 |
| 0.3805141 | 0.8827992 | 0.5769283 | 0.8612326 | -0.5262797 | -0.3016326 | -0.7729720 | -0.5573386 | -0.3059767 | 72 |
| 0.4233029 | 0.8202211 | -0.3420725 | 0.2861647 | -0.8249789 | 0.0473638 | 0.0761516 | -0.7020124 | -0.3059767 | 72 |
| 0.0809930 | 0.9453774 | 0.1940113 | 1.4363005 | 1.1912411 | 0.5708584 | 0.0478474 | -0.6537878 | -0.4399811 | 72 |
| -0.4324719 | 0.1944393 | 0.1940113 | -0.8639710 | -0.6756293 | -0.5633799 | 1.4630533 | -0.6055632 | -0.4399811 | 72 |
| -0.9031480 | -0.9945460 | 0.5769283 | -0.2889031 | -0.3769300 | -0.8251272 | -1.5371832 | -0.5091140 | -0.4399811 | 71 |
| 0.5088804 | 1.3208464 | 0.2705947 | 0.2861647 | -0.8249789 | 1.0943531 | -0.2634979 | -0.7020124 | -0.5739855 | 71 |
| 2.3915849 | 0.5073302 | -1.1079066 | 0.2861647 | 0.4444930 | -0.0835099 | -0.4050184 | -0.3644402 | 0.2300409 | 76 |
| 0.4233029 | -0.0558734 | 0.5003449 | 0.2861647 | 0.2951433 | 1.2252267 | -0.3767143 | -0.7020124 | -0.4399811 | 75 |
| 0.4233029 | 0.4447520 | -1.0313232 | 1.4363005 | -0.8249789 | -1.4794955 | 1.2932286 | -0.5091140 | -0.1719723 | 75 |
| 0.3377254 | 0.6324865 | -0.5718228 | -0.2889031 | -0.3769300 | -1.7848674 | -0.2918020 | -0.6055632 | -0.3059767 | 74 |
| -0.7747818 | -1.8706404 | 0.9598453 | -0.8639710 | -0.8249789 | -0.9996255 | 0.0761516 | -0.7020124 | 0.2300409 | 73 |
| -0.0045845 | 1.0705337 | 0.0408445 | 1.4363005 | -0.7503041 | -0.1271344 | -0.7163637 | -0.6055632 | -0.4399811 | 73 |
| 0.4233029 | 0.6950647 | -0.1123223 | -0.8639710 | -0.8249789 | 1.4433495 | 0.6422339 | 0.0695812 | 1.0340673 | 71 |
| 2.8622610 | 0.8202211 | -0.7249896 | 0.8612326 | -1.0490034 | 1.0507285 | -0.1219773 | -0.6537878 | -0.4399811 | 71 |
| 1.1935002 | 1.3834246 | -0.1123223 | 0.8612326 | -0.6009545 | 1.5742231 | 0.9252751 | -0.5573386 | -0.0379679 | 70 |
| 0.7656128 | 0.3195957 | 0.1940113 | 0.8612326 | 1.1912411 | 0.0909884 | 0.1327598 | -0.5091140 | 1.0340673 | 70 |
| 0.2949367 | -0.1810297 | 0.1174279 | -0.8639710 | 1.4152656 | -1.1741237 | 0.7554504 | 0.5036026 | -0.5739855 | 70 |
| -0.6464156 | 0.6324865 | 0.1174279 | 0.2861647 | -0.1529056 | 0.4836093 | -0.8861884 | -0.6055632 | 0.9000629 | 75 |
| 0.3805141 | 0.0692830 | 0.2705947 | -0.2889031 | 0.6685174 | 0.2218620 | -0.3201061 | -0.3162156 | -0.5739855 | 73 |
| -0.7747818 | -0.8068114 | 0.3471781 | -0.8639710 | 0.9672167 | -0.4325063 | -0.9144926 | 0.1178058 | 0.0960365 | 73 |
| 0.0809930 | -0.3061861 | -0.4952393 | 0.2861647 | -0.5262797 | -1.3922464 | 1.5196616 | -0.1715418 | -0.7079900 | 72 |
| -1.3310354 | 0.4447520 | -0.6484062 | 0.2861647 | -0.5262797 | -0.8251272 | 0.1610639 | 0.7929502 | -0.5739855 | 71 |
| 1.7925425 | 1.5085810 | -1.1844900 | 0.8612326 | 1.1165663 | -0.3452572 | -0.6597555 | -0.7020124 | 0.6320541 | 71 |
| -0.9887255 | 0.0692830 | 2.0320130 | 0.8612326 | -0.9743286 | -0.6070045 | -0.7163637 | -0.2679910 | -0.1719723 | 70 |
| -0.8175705 | 0.0067048 | -1.3376568 | 0.8612326 | -0.7503041 | -1.2177482 | 0.3025845 | -0.6537878 | -0.7079900 | 76 |
| 3.0762047 | 0.1944393 | -2.7927415 | -0.2889031 | 3.0581116 | -0.8251272 | 0.1610639 | -0.7020124 | 0.0960365 | 76 |
| 0.4233029 | 0.9453774 | -1.7971572 | 0.8612326 | 1.1912411 | -0.7378781 | -0.3484102 | -0.5573386 | -0.0379679 | 76 |
| -0.7747818 | 0.3821738 | 1.8022628 | 0.8612326 | -0.4516048 | 0.3091111 | -0.0087608 | -0.4126648 | 0.0960365 | 75 |
| 0.5088804 | -0.0558734 | 0.3471781 | 0.2861647 | 0.1457937 | -1.0432500 | -0.2918020 | -0.6537878 | -0.4399811 | 75 |
| -0.9887255 | -0.6190769 | 1.2661790 | -0.8639710 | 3.4314856 | 1.4433495 | -0.1502814 | -0.6537878 | -0.1719723 | 74 |
| 0.5088804 | 0.5073302 | -1.7205738 | 0.2861647 | -0.2275804 | -0.0835099 | 0.8969710 | -0.6537878 | -0.4399811 | 73 |
| 0.1237817 | -0.7442333 | 1.1895955 | -0.8639710 | 0.0711189 | 0.1346129 | 0.0761516 | 0.0695812 | -0.5739855 | 73 |
| 1.7069650 | -1.6203277 | -1.2610734 | -0.2889031 | -0.6756293 | -0.9996255 | 1.9725274 | 0.9376240 | -0.7079900 | 73 |
| -1.6733453 | 0.4447520 | 0.1174279 | 0.2861647 | 1.0418915 | 1.3561004 | -1.1692296 | 1.1305224 | 1.4360806 | 72 |
| 0.7656128 | 0.0067048 | 0.8066785 | 0.2861647 | -0.3769300 | 0.1782375 | -0.3201061 | -0.6537878 | 0.6320541 | 72 |
| -1.3738241 | 0.1318611 | -0.3420725 | 0.8612326 | 1.1165663 | 0.0473638 | -0.9144926 | 0.4553780 | 0.9000629 | 72 |
| 2.4771623 | 0.5699084 | -0.1123223 | -0.8639710 | -0.3022552 | 1.7050968 | -0.2068896 | 0.3107042 | 1.0340673 | 72 |
| -0.9887255 | 0.5699084 | -0.4186559 | 0.2861647 | -0.7503041 | -0.3888817 | -0.4899308 | -0.0750926 | -0.3059767 | 72 |
| 0.0809930 | -0.2436079 | 1.3427624 | 0.8612326 | -0.8996538 | 1.8359704 | 0.2742804 | 0.7447256 | -0.4399811 | 71 |
| -1.0743030 | 0.0067048 | 0.2705947 | 0.8612326 | 1.9379893 | -0.3016326 | -0.0936732 | -0.6537878 | -0.0379679 | 70 |
| 0.0809930 | 0.5699084 | -0.2654891 | -1.4390389 | 0.5938426 | 0.0037392 | -0.2068896 | -0.6537878 | -0.4399811 | 76 |
| -0.0045845 | -0.3687642 | 1.1130121 | -1.4390389 | -0.6009545 | -0.5633799 | 1.0667957 | -0.4608894 | 0.2300409 | 74 |
| -0.7747818 | 0.5699084 | 0.5003449 | -0.8639710 | 0.9672167 | -0.7815027 | 0.0478474 | -0.5091140 | -0.3059767 | 74 |
| -1.0743030 | 0.5699084 | -0.2654891 | 0.2861647 | -0.1529056 | -0.0835099 | -1.1409255 | -0.3644402 | -0.5739855 | 73 |
| 0.2521479 | -0.1184515 | 2.1851798 | -0.2889031 | 2.2366886 | 1.3997249 | -1.2541420 | -0.2197664 | 0.2300409 | 73 |
| -0.4324719 | 1.0705337 | 0.8066785 | 0.8612326 | 0.3698181 | 1.0071040 | -1.6787037 | -0.5091140 | -0.1719723 | 73 |
| 0.2521479 | -0.5564988 | 0.0408445 | 0.2861647 | -0.3022552 | -0.5197554 | 0.2459763 | 0.5036026 | -0.0379679 | 72 |
| -1.2026692 | -0.7442333 | 0.1174279 | -2.0141068 | -0.4516048 | -0.9560009 | -0.8861884 | -0.2197664 | -0.7079900 | 72 |
| -0.3041057 | -1.3074368 | 1.2661790 | -0.8639710 | 1.1912411 | -0.2143835 | -0.6880596 | -0.6537878 | 0.4980497 | 71 |
| 1.9209087 | -0.0558734 | -0.1123223 | -0.2889031 | -0.5262797 | -0.0835099 | -0.2918020 | -0.6537878 | 0.9000629 | 71 |
| -1.1170917 | 0.5699084 | -0.6484062 | 0.8612326 | 4.1782338 | 0.5708584 | -0.3201061 | -0.6537878 | 1.8380938 | 70 |
| 0.0809930 | 1.0705337 | 0.8832619 | 0.2861647 | -0.6009545 | -0.5197554 | -0.2068896 | -0.6537878 | -0.5739855 | 70 |
| 3.1617822 | 0.1318611 | 0.1940113 | -0.2889031 | -0.8996538 | 0.9198548 | -0.4333226 | -0.6537878 | 2.5081158 | 76 |
| -0.5180494 | -0.4313424 | -0.2654891 | -1.4390389 | -0.4516048 | -1.6976183 | -0.0936732 | -0.1233172 | -0.3059767 | 76 |
| -0.7747818 | -0.4313424 | 0.0408445 | -0.8639710 | -0.6756293 | -1.1741237 | -0.0936732 | -0.6055632 | -0.7079900 | 76 |
| -1.4166129 | -0.5564988 | -2.4098245 | -0.2889031 | -0.1529056 | 0.3527357 | -0.2634979 | 0.8893994 | 3.9821643 | 75 |
| -0.6036269 | 0.5073302 | 0.7300951 | 0.8612326 | -0.9743286 | 0.2218620 | -1.1692296 | -0.2197664 | -0.5739855 | 75 |
| -0.7747818 | 0.0692830 | 0.3471781 | -0.2889031 | -0.9743286 | -0.4761308 | 0.2459763 | -0.0268680 | 0.3640453 | 72 |
| -0.8175705 | -2.1209531 | -0.8781564 | -3.1642426 | -0.5262797 | -1.2613728 | 1.3498369 | 1.6609930 | -0.7079900 | 72 |
| -1.0315142 | 1.2582683 | -0.8781564 | 1.4363005 | -0.6009545 | 0.4399848 | -1.0277090 | 0.0213566 | 0.0960365 | 71 |
| -1.2026692 | -0.7442333 | -0.1123223 | -0.8639710 | -0.6009545 | 0.7017321 | -0.8861884 | 0.2624796 | 0.3640453 | 71 |
| -0.7747818 | -1.2448587 | 0.4237615 | 0.2861647 | -0.6756293 | 0.0037392 | -0.4050184 | -0.3162156 | -0.7079900 | 71 |
| 0.2949367 | 0.0692830 | 2.1851798 | -0.2889031 | 1.2659159 | -0.2580081 | -0.0370649 | -0.3644402 | 0.3640453 | 70 |
| -0.5608381 | 1.0079556 | -1.8737406 | 0.8612326 | 1.3405908 | -0.2143835 | -0.6880596 | 0.1178058 | 3.0441335 | 70 |
| 0.2521479 | 1.0705337 | 0.3471781 | 0.8612326 | -0.2275804 | -0.1271344 | -0.9427967 | -0.5091140 | -0.3059767 | 70 |
| 0.2949367 | 1.0079556 | 0.5003449 | 1.4363005 | -0.4516048 | 0.8762303 | -0.6314514 | -0.5573386 | -0.5739855 | 70 |
| -0.8603593 | -0.2436079 | 0.1940113 | 1.4363005 | -0.4516048 | 0.4836093 | -0.6031473 | -0.5573386 | -0.1719723 | 76 |
| -0.6464156 | -1.5577495 | 0.4237615 | -1.4390389 | -0.9743286 | -0.1707590 | -0.5748432 | -0.7020124 | -0.0379679 | 76 |
| -0.7747818 | 0.6324865 | 0.1174279 | -0.2889031 | -0.9743286 | -0.2580081 | -0.1785855 | -0.5573386 | -0.0379679 | 76 |
| -1.2026692 | -0.8068114 | 0.1174279 | -1.4390389 | -0.4516048 | -0.3016326 | -0.9144926 | -0.1715418 | -0.4399811 | 76 |
| 0.4233029 | 0.0067048 | 0.9598453 | -0.2889031 | -0.3769300 | -0.0398853 | -0.3767143 | 0.2624796 | 0.4980497 | 75 |
| -0.4752606 | 0.5699084 | -0.1123223 | 0.8612326 | -0.6009545 | -0.8687518 | -0.2918020 | -0.5091140 | -0.3059767 | 74 |
| -0.9031480 | -0.4939206 | 0.5003449 | -0.8639710 | -0.7503041 | -0.6506290 | 1.5196616 | -0.2197664 | -0.0379679 | 74 |
| 0.0809930 | -0.8693896 | -0.9547398 | 0.2861647 | -0.5262797 | -1.3486219 | 3.2745169 | -0.5573386 | -0.7079900 | 72 |
| -1.0743030 | 0.6324865 | -0.0357389 | 0.2861647 | -0.7503041 | 1.7050968 | -0.8012761 | -0.6055632 | 3.3121423 | 72 |
| 2.1348524 | -0.8693896 | -1.4142402 | -0.2889031 | -0.6009545 | -1.4358710 | -1.1409255 | 0.4553780 | -0.4399811 | 72 |
| -1.2026692 | -0.7442333 | -0.1123223 | -0.8639710 | -0.6009545 | 0.7017321 | -0.8861884 | 0.2624796 | 0.3640453 | 71 |
| -1.0315142 | 1.0079556 | 0.7300951 | 1.4363005 | -1.0490034 | 0.3963602 | -1.2541420 | 0.1178058 | -0.4399811 | 76 |
| -0.6036269 | -0.3061861 | -0.2654891 | -0.8639710 | 0.1457937 | -1.5667446 | -1.3956626 | -0.6537878 | -0.7079900 | 75 |
| -0.9887255 | 0.5073302 | 0.8066785 | -0.2889031 | -0.8249789 | 0.3963602 | -0.5465390 | 0.3107042 | 0.4980497 | 75 |
| 0.7656128 | 0.3195957 | -0.8781564 | 0.2861647 | -0.6756293 | 0.5708584 | 0.9818833 | -0.3644402 | 3.4461467 | 70 |
| -0.9031480 | 1.1331119 | 0.1940113 | 0.8612326 | -0.7503041 | 1.2252267 | -0.2351937 | 0.2142550 | -0.5739855 | 83 |
| 0.0382042 | 1.3208464 | 0.4237615 | 0.8612326 | -0.7503041 | 0.8326057 | 1.9725274 | -0.7020124 | -0.5739855 | 79 |
| -0.3896832 | 0.8202211 | -0.0357389 | 0.2861647 | -0.6756293 | -0.3888817 | -0.1219773 | -0.6055632 | -0.5739855 | 78 |
| -2.2723877 | 0.9453774 | 0.6535117 | 0.2861647 | -0.5262797 | 0.0909884 | -0.3201061 | -0.5091140 | 0.7660585 | 77 |
| 0.0809930 | 0.4447520 | 0.8066785 | 0.8612326 | -1.0490034 | 1.3124758 | -0.6314514 | -0.7020124 | 0.0960365 | 77 |
| 0.8939790 | -1.8706404 | -1.7205738 | -0.8639710 | -0.5262797 | -1.5667446 | 1.6894863 | 0.7929502 | -0.7079900 | 81 |
| 0.7228241 | -2.3712658 | -0.6484062 | -1.4390389 | -0.6009545 | -1.0432500 | 1.6894863 | 3.5899770 | -0.7079900 | 80 |
| 0.4660916 | 1.4460028 | 0.1940113 | 1.4363005 | 0.8925419 | 0.0473638 | 0.7554504 | 0.2142550 | 1.0340673 | 79 |
| 0.3377254 | 0.5073302 | -0.3420725 | 0.8612326 | -0.3769300 | -0.9560009 | 0.5573216 | -0.3644402 | 0.2300409 | 80 |
| 1.1079227 | -0.0558734 | 0.6535117 | 0.8612326 | 0.9672167 | 1.5305986 | 0.2176721 | -0.4608894 | -0.5739855 | 79 |
| 0.9367677 | -0.7442333 | -1.4908236 | -0.8639710 | -0.7503041 | -1.3922464 | 1.5196616 | -0.4608894 | -0.7079900 | 79 |
| -0.3468944 | 0.2570175 | 0.7300951 | 0.2861647 | 0.8178670 | 0.1346129 | 0.1610639 | -0.6055632 | -0.5739855 | 79 |
| 0.2949367 | 0.5073302 | 0.6535117 | 2.0113684 | -0.8996538 | 1.6178477 | -0.7163637 | -0.6537878 | -0.1719723 | 78 |
| -1.7589228 | -1.4951713 | -0.1889057 | -0.8639710 | -1.1236782 | 0.7017321 | -0.1502814 | -0.7020124 | -0.5739855 | 77 |
| 0.2521479 | 0.4447520 | -0.8781564 | -0.2889031 | -0.1529056 | 0.5272339 | 0.6139298 | -0.7020124 | -0.4399811 | 85 |
| 0.1237817 | -0.3061861 | -0.3420725 | -0.2889031 | -0.4516048 | -0.9996255 | 0.5573216 | 0.4071534 | -0.4399811 | 81 |
| -0.9887255 | -0.9945460 | 1.0364287 | -0.8639710 | -0.3769300 | -0.3452572 | 0.5007133 | 0.9858486 | -0.4399811 | 81 |
| -1.4166129 | 1.0079556 | -1.7205738 | 0.8612326 | -0.4516048 | 0.4836093 | -0.6314514 | -0.3644402 | -0.3059767 | 80 |
| -1.6305566 | -0.5564988 | -0.2654891 | -0.2889031 | 1.4152656 | -0.1707590 | 0.1610639 | -0.6537878 | -0.7079900 | 78 |
| 0.4660916 | 0.8202211 | -0.2654891 | 1.4363005 | -0.4516048 | -0.3452572 | -0.3484102 | -0.1715418 | -0.4399811 | 78 |
| -0.0901620 | 0.6324865 | 1.1130121 | -2.0141068 | -0.6009545 | 1.4869740 | 0.6422339 | -0.5091140 | -0.5739855 | 77 |
| 0.0809930 | -2.8093130 | -1.7205738 | -0.8639710 | -0.3769300 | -0.8251272 | 1.2649245 | 3.7346508 | -0.7079900 | 86 |
| -1.4594016 | -1.8080622 | -1.7205738 | -0.2889031 | -0.5262797 | -1.4794955 | -1.4805749 | 2.1432390 | -0.5739855 | 85 |
| 0.4233029 | 0.9453774 | -0.3420725 | 0.8612326 | -0.3022552 | 0.9634794 | -0.0653690 | -0.6055632 | -0.7079900 | 80 |
| 1.7925425 | 0.6950647 | 0.0408445 | 0.2861647 | -0.6756293 | 1.9232196 | 0.2176721 | 0.3589288 | 1.3020762 | 79 |
| 1.1079227 | 0.3195957 | 0.9598453 | 0.2861647 | 0.9672167 | 1.7923459 | -0.0087608 | -0.5091140 | -0.5739855 | 79 |
| 1.7925425 | 0.5699084 | 0.0408445 | 0.2861647 | 1.1165663 | 1.8359704 | -0.1785855 | -0.4126648 | 1.1680718 | 78 |
| 0.2949367 | 0.4447520 | -0.7249896 | 0.2861647 | -0.7503041 | -0.2580081 | 0.8403627 | -0.5573386 | -0.5739855 | 78 |
| -0.0901620 | -0.1184515 | 1.7256794 | 0.2861647 | 1.0418915 | 1.8359704 | -1.3107502 | -0.5091140 | -0.5739855 | 77 |
| -1.8445003 | -1.0571241 | -1.0313232 | -0.8639710 | -0.6009545 | -1.2177482 | 0.7837545 | 1.1787470 | -0.3059767 | 77 |
| -1.5449791 | -1.3700150 | 0.2705947 | -0.2889031 | -0.7503041 | -0.1271344 | 0.3874969 | 1.8538914 | -0.5739855 | 84 |
| -0.7747818 | -2.0583749 | -0.2654891 | -2.5891747 | 0.7431922 | 0.0037392 | 0.2459763 | 3.3488540 | 4.7861908 | 83 |
| -0.1329507 | -0.6190769 | 1.6490960 | -0.8639710 | -1.1236782 | 1.7923459 | 1.2649245 | -0.6537878 | -0.0379679 | 82 |
| 0.0809930 | 0.5073302 | 0.8066785 | -0.8639710 | -0.6756293 | 1.5742231 | 0.2742804 | -0.4126648 | -0.7079900 | 81 |
| -0.7747818 | 0.3195957 | 0.0408445 | 0.8612326 | 0.8925419 | -0.4761308 | -0.6880596 | -0.3162156 | -0.4399811 | 81 |
| 3.7608245 | -2.7467348 | -0.4952393 | -0.2889031 | -0.3769300 | -0.7378781 | -1.0560132 | 3.5417524 | -0.4399811 | 80 |
| 0.0809930 | -2.0583749 | 0.1174279 | -0.2889031 | -0.6009545 | -0.3452572 | 0.7554504 | 1.1787470 | -0.7079900 | 79 |
| 0.2949367 | -0.8693896 | 1.5725126 | -1.4390389 | -0.7503041 | 1.1816022 | -0.5465390 | 2.1432390 | 0.2300409 | 79 |
| -1.0315142 | -0.0558734 | 0.3471781 | 0.2861647 | -0.3769300 | 0.9634794 | -0.9427967 | -0.6055632 | -0.4399811 | 77 |
| 0.0809930 | -1.2448587 | 1.4193458 | -2.0141068 | -0.7503041 | -0.3016326 | -0.4899308 | -0.6537878 | -0.5739855 | 77 |
| 0.0809930 | -1.1822805 | 0.1940113 | -1.4390389 | -0.4516048 | -0.6506290 | 0.0478474 | 3.4453032 | -0.5739855 | 77 |
| 0.1665704 | 1.0079556 | 0.7300951 | 0.8612326 | 1.5646152 | -0.7815027 | -0.3484102 | -0.5091140 | -0.1719723 | 77 |
| 1.1935002 | 0.1318611 | -0.4952393 | -0.2889031 | 0.6685174 | -1.0432500 | -0.3484102 | -0.7020124 | -0.4399811 | 77 |
| -0.2185282 | -0.9319678 | -0.4952393 | -0.8639710 | -0.3022552 | -1.6539937 | 0.4724092 | -0.3644402 | -0.4399811 | 84 |
| 0.0382042 | 0.2570175 | 0.9598453 | 0.2861647 | -1.0490034 | 1.6614722 | -0.1785855 | 0.0213566 | -0.4399811 | 81 |
| 1.7925425 | 1.0079556 | -0.8781564 | 0.2861647 | -0.3022552 | -0.4325063 | -0.4899308 | 0.0695812 | 1.1680718 | 81 |
| 0.6800353 | -1.1822805 | -1.1079066 | -0.2889031 | 2.0126641 | -1.5231201 | -0.8295802 | -0.1715418 | -0.5739855 | 81 |
| 0.2521479 | 1.3834246 | -0.1123223 | 1.4363005 | -0.2275804 | -0.2143835 | -0.0936732 | 0.2624796 | -0.7079900 | 80 |
| 0.0809930 | -1.4951713 | 0.9598453 | -0.2889031 | 1.5646152 | -0.7815027 | 0.3308886 | 0.6965010 | -0.4399811 | 80 |
| -0.2185282 | 0.7576429 | 0.1940113 | -0.2889031 | 0.0711189 | 0.7889812 | 0.3025845 | -0.6537878 | -0.1719723 | 79 |
| -0.5608381 | -0.7442333 | 1.1895955 | -0.2889031 | -0.8996538 | 0.3091111 | -0.3201061 | -0.2197664 | -0.5739855 | 79 |
| -0.4324719 | -1.1197023 | -0.1123223 | -0.8639710 | -0.0035559 | -0.4325063 | -0.0087608 | 1.2269716 | -0.5739855 | 79 |
| -1.1170917 | -1.1197023 | 0.6535117 | -0.2889031 | -0.6756293 | 0.1782375 | -0.1785855 | -0.3162156 | 0.0960365 | 78 |
| 1.3646551 | 0.2570175 | 0.8066785 | -0.8639710 | -0.9743286 | 1.0507285 | -0.5182349 | -0.6055632 | -0.1719723 | 78 |
| -0.2613169 | -0.2436079 | 0.8832619 | -0.8639710 | -0.1529056 | -0.6070045 | 0.7271463 | -0.6055632 | 0.2300409 | 78 |
| -0.6892043 | 0.0067048 | 0.1174279 | -0.8639710 | -0.7503041 | 0.7453566 | -0.7729720 | -0.5091140 | 0.3640453 | 78 |
| -0.7747818 | -0.5564988 | 0.6535117 | 0.8612326 | -0.5262797 | -0.0398853 | -1.2258379 | 1.1305224 | -0.3059767 | 78 |
| 0.3377254 | 0.2570175 | 0.9598453 | 0.8612326 | -0.5262797 | 1.2252267 | 0.5856257 | -0.6055632 | 0.0960365 | 77 |
| -0.9459368 | -0.3687642 | 1.4193458 | 1.4363005 | -0.6756293 | 1.7923459 | 0.2459763 | 2.0950144 | -0.0379679 | 87 |
| -1.5021904 | 0.6950647 | -1.5674070 | 0.8612326 | -0.5262797 | 1.0071040 | -1.0560132 | -0.1715418 | 3.4461467 | 84 |
| 0.6800353 | -0.9319678 | 1.5725126 | 0.2861647 | -0.6756293 | 1.4869740 | -0.5182349 | -0.6537878 | -0.5739855 | 83 |
| -0.7747818 | 0.0692830 | 0.0408445 | 0.2861647 | -0.3022552 | -1.5667446 | -0.5465390 | -0.2679910 | -0.7079900 | 82 |
| -1.0743030 | 0.6950647 | 0.9598453 | -0.2889031 | -0.8249789 | 0.3091111 | 1.2366204 | 1.3716454 | 0.4980497 | 81 |
| 1.2790777 | -0.3061861 | -0.6484062 | 0.2861647 | -0.5262797 | -0.6942536 | -0.4616267 | -0.0750926 | -0.0379679 | 81 |
| -0.6892043 | -0.5564988 | 0.3471781 | 0.2861647 | -0.4516048 | 0.0909884 | -0.4333226 | 0.5036026 | -0.3059767 | 80 |
| -0.7747818 | -1.1197023 | -0.0357389 | -0.8639710 | -0.6756293 | 0.0037392 | -0.1219773 | -0.1233172 | -0.1719723 | 79 |
| -0.6464156 | -0.3687642 | -0.2654891 | -0.8639710 | -0.0782308 | -0.0398853 | 0.6422339 | -0.5573386 | -0.5739855 | 78 |
| -0.0045845 | -1.6203277 | 0.7300951 | -1.4390389 | 1.1165663 | -1.8721165 | 0.5573216 | -0.4126648 | -0.3059767 | 78 |
| 0.0809930 | 1.2582683 | 1.3427624 | 1.4363005 | -0.1529056 | 0.0909884 | -0.1785855 | -0.2679910 | -0.5739855 | 78 |
| -1.9300778 | -1.9957967 | -0.4952393 | -0.8639710 | -0.5262797 | -0.8687518 | 0.1327598 | 1.6609930 | -0.7079900 | 77 |
| 0.2521479 | 0.9453774 | 1.1130121 | 1.4363005 | -0.7503041 | 0.3091111 | -0.2918020 | -0.5091140 | 0.3640453 | 86 |
| -0.5608381 | -1.1822805 | 0.9598453 | 0.8612326 | -0.3769300 | -0.5633799 | -0.2351937 | 0.1660304 | -0.7079900 | 85 |
| -0.0901620 | -0.0558734 | 0.8832619 | -0.2889031 | -0.2275804 | 0.3963602 | -0.4616267 | -0.7020124 | -0.1719723 | 84 |
| 0.1665704 | 0.6324865 | 1.7256794 | 0.8612326 | -0.7503041 | 1.9668441 | -0.7446679 | 1.1305224 | -0.3059767 | 84 |
| 0.1665704 | 0.0067048 | 0.4237615 | -0.2889031 | 0.8178670 | -0.7378781 | -0.5465390 | -0.5091140 | -0.3059767 | 84 |
| 0.9367677 | 0.0067048 | -0.1889057 | -0.8639710 | -0.3769300 | -0.1271344 | -0.4050184 | -0.4126648 | -0.1719723 | 83 |
| 0.2949367 | 0.5699084 | 0.5769283 | 0.2861647 | 0.1457937 | -0.9123764 | -0.7729720 | -0.1233172 | -0.5739855 | 83 |
| -0.7747818 | -0.6190769 | 1.0364287 | 0.2861647 | -0.3769300 | 1.2252267 | -0.4616267 | -0.1715418 | -0.4399811 | 83 |
| -0.0045845 | -0.2436079 | -0.4186559 | -0.2889031 | 1.1912411 | -0.0398853 | 0.3591927 | -0.6537878 | -0.4399811 | 82 |
| 2.3060074 | -0.4313424 | -1.0313232 | 0.2861647 | -0.7503041 | -0.6070045 | -0.0370649 | -0.3644402 | -0.4399811 | 82 |
| -0.6464156 | 0.0067048 | 0.1174279 | 0.8612326 | -0.7503041 | 0.9634794 | -1.4239667 | -0.5573386 | -0.4399811 | 82 |
| 1.0223452 | 0.3195957 | -0.9547398 | 1.4363005 | -0.0782308 | -0.9996255 | -1.2541420 | 0.5518272 | 0.0960365 | 82 |
| -1.1170917 | 0.6324865 | 0.0408445 | 0.8612326 | -0.0782308 | 0.1346129 | -0.4050184 | 0.2142550 | -0.4399811 | 80 |
| -0.3468944 | -0.6816551 | 1.1895955 | 0.2861647 | -0.6009545 | -0.4325063 | -0.7163637 | -0.3162156 | -0.3059767 | 79 |
| -0.5608381 | 0.9453774 | 0.4237615 | 0.2861647 | -1.1983530 | 1.7923459 | -0.2351937 | -0.6055632 | 0.2300409 | 78 |
| -0.5608381 | 0.1318611 | 0.7300951 | 0.2861647 | 0.8925419 | 0.4399848 | -1.0277090 | -0.7020124 | -0.5739855 | 77 |
| -0.3041057 | -0.1810297 | -1.2610734 | 0.2861647 | -0.7503041 | -0.7378781 | -0.6597555 | 0.0213566 | 0.0960365 | 93 |
| 0.3805141 | -1.5577495 | -0.1889057 | -0.2889031 | -0.3022552 | -0.0835099 | -1.1692296 | 1.9985652 | -0.7079900 | 92 |
| -0.7747818 | 0.5073302 | 0.3471781 | 1.4363005 | -0.0782308 | 0.6581075 | -0.3201061 | 0.2142550 | -0.4399811 | 90 |
| 1.7069650 | -0.5564988 | 0.5769283 | 1.4363005 | -0.4516048 | -0.4761308 | -1.4805749 | -0.1715418 | 0.3640453 | 90 |
| -0.9459368 | 1.1331119 | 0.1940113 | 1.4363005 | 4.1782338 | -0.1707590 | -1.8768326 | 0.7447256 | -0.3059767 | 90 |
| -1.0315142 | -1.3074368 | 1.1895955 | -0.2889031 | -0.8996538 | 1.5305986 | -1.1975337 | -0.1715418 | -0.1719723 | 87 |
| -1.0743030 | -0.3687642 | -0.0357389 | -1.4390389 | -0.7503041 | 0.1346129 | -0.7729720 | -0.1233172 | 1.4360806 | 86 |
| 0.8511903 | 0.1318611 | 1.4959292 | 0.2861647 | -0.5262797 | 1.8795950 | -1.2541420 | -0.5091140 | -0.3059767 | 86 |
| -0.8175705 | 1.0705337 | 0.1940113 | 1.4363005 | -0.7503041 | -1.2613728 | -1.3390543 | 0.7447256 | 0.9000629 | 86 |
| -0.6892043 | -0.1810297 | 1.1895955 | 0.2861647 | -0.6756293 | 1.3997249 | -0.1502814 | -0.2679910 | -0.5739855 | 85 |
| 0.7656128 | -1.9332186 | 1.9554296 | -2.0141068 | -0.8996538 | -0.7378781 | -0.7729720 | -0.4608894 | 0.6320541 | 84 |
| 0.2949367 | 0.9453774 | 0.6535117 | 0.2861647 | -0.5262797 | 0.0037392 | -2.1881779 | 1.6127684 | -0.7079900 | 82 |
| 0.2949367 | -1.0571241 | 1.5725126 | -0.8639710 | -0.3769300 | -0.3016326 | 0.3025845 | 2.0467898 | -0.4399811 | 81 |
| -0.6036269 | -0.0558734 | -0.4186559 | -0.2889031 | 1.1912411 | 0.0037392 | -0.2068896 | 0.2624796 | -0.7079900 | 81 |
| 1.4502326 | 0.6950647 | 0.5769283 | 0.2861647 | 1.3405908 | 1.7050968 | -1.7919202 | -0.0750926 | -0.4399811 | 78 |
| -0.9459368 | 0.5699084 | 1.4959292 | 0.8612326 | -0.8996538 | 0.4399848 | -0.3484102 | 0.0695812 | 1.9720982 | 77 |
By using k-fold cross validation, it will help with imbalanced data
and making sure that our training sets have an accurate representation
on our overall data. We will again stratify on the response variable,
pop.
set.seed(100)
# Creating folds
spotify_folds <- vfold_cv(spotify_train, v = 5, strata = pop)
With a 5-fold cross validation and a little over 600 observations in total, there will be approximately 120 observations in each fold. This should suffice with the amount of total observations we have.
Linear Regression, Ridge Regression, K-Nearest Neighbors, Elastic Net, Random Forest, Support Vector Machine, and Boosted Trees will be the models I will be looking at.
For this process, we will follow these steps to fit them:
# Linear Regression
lm_model <- linear_reg() %>%
set_engine("lm")
# Ridge Regression
# Tuning penalty and letting mixture be set to 0
ridge_model <- linear_reg(mixture = 0,
penalty = tune()) %>%
set_mode("regression") %>%
set_engine("glmnet")
# K-Nearest Neighbors
# Tuning neighbors
knn_model <- nearest_neighbor(neighbors = tune()) %>%
set_mode("regression") %>%
set_engine("kknn")
# Elastic Net
# Tuning penalty and mixture
en_model <- linear_reg(penalty = tune(),
mixture = tune()) %>%
set_mode("regression") %>%
set_engine("glmnet")
# Random Forest
# Tuning mtry, trees, and min_n
rf_model <- rand_forest(mtry = tune(),
trees = tune(),
min_n = tune()) %>%
set_engine("ranger", importance = "impurity") %>%
set_mode("regression")
# Support Vector Machine
# Tuning cost and letting degree be set to 1
svm_model <- svm_poly(degree = 1, cost = tune()) %>%
set_engine("kernlab") %>%
set_mode("regression")
# Boosted Trees
# Tuning mtry, trees, and learn_rate
bt_model <- boost_tree(mtry = tune(),
trees = tune(),
learn_rate = tune()) %>%
set_engine("xgboost") %>%
set_mode("regression")
# Linear Regression
lm_workflow <- workflow() %>%
add_model(lm_model) %>%
add_recipe(spotify_recipe)
# Ridge Regression
ridge_workflow <- workflow() %>%
add_model(ridge_model) %>%
add_recipe(spotify_recipe)
# K-Nearest Neighbors
knn_workflow <- workflow() %>%
add_model(knn_model) %>%
add_recipe(spotify_recipe)
# Elastic Net
en_workflow <- workflow() %>%
add_model(en_model) %>%
add_recipe(spotify_recipe)
# Random Forest
rf_workflow <- workflow() %>%
add_model(rf_model) %>%
add_recipe(spotify_recipe)
# Support Vector Model
svm_workflow <- workflow() %>%
add_model(svm_model) %>%
add_recipe(spotify_recipe)
# Boosted Trees
bt_workflow <- workflow() %>%
add_model(bt_model) %>%
add_recipe(spotify_recipe)
# Linear Regression (N/A because there is no tuning parameters)
# Ridge Regression
ridge_grid <- grid_regular(penalty(range = c(0.1,5)), levels = 50)
# K-Nearest Neighbors
knn_grid <- grid_regular(neighbors(range = c(1,15)), levels = 5)
# Elastic Net
en_grid <- grid_regular(penalty(range = c(0.1, 5)), mixture(range = c(0,1)), levels = 10)
# Random Forest
rf_grid <- grid_regular(mtry(range = c(1, 9)), trees(range = c(200,1000)), min_n(range = c(5,20)), levels = 8)
# Support Vector Machine
svm_grid <- grid_regular(cost(), levels = 8)
# Boosted Trees
bt_grid <- grid_regular(mtry(range = c(1, 6)),
trees(range = c(200, 600)),
learn_rate(range = c(-10, -1)),
levels = 5)
# Linear Regression (N/A because there is no tuning parameters)
# Ridge Regression
ridge_tune <- tune_grid(ridge_workflow,
resamples = spotify_folds,
grid = ridge_grid)
# K-Nearest Neighbors
knn_tune <- tune_grid(knn_workflow,
resamples = spotify_folds,
grid = knn_grid)
# Elastic Net
en_tune <- tune_grid(en_workflow,
resamples = spotify_folds,
grid = en_grid)
# Random Forest
rf_tune <- tune_grid(rf_workflow,
resamples = spotify_folds,
grid = rf_grid)
# Support Vector Machine
svm_tune <- tune_grid(svm_workflow,
resamples = spotify_folds,
grid = svm_grid)
# Boosted Trees
bt_tune <- tune_grid(bt_workflow,
resamples = spotify_folds,
grid = bt_grid)
# Linear Regression (N/A because there is no tuning parameters)
# Ridge Regression
save(ridge_tune, file = "ridge_tune.rda")
# K-Nearest Neighbors
save(knn_tune, file = "knn_tune.rda")
# Elastic Net
save(en_tune, file = "en_tune.rda")
# Random Forest
save(rf_tune, file = "rf_tune.rda")
# Support Vector Machine
save(svm_tune, file = "svm_tune.rda")
# Boosted Trees
save(bt_tune, file = "bt_tune.rda")
load("ridge_tune.rda")
load("knn_tune.rda")
load("en_tune.rda")
load("rf_tune.rda")
load("svm_tune.rda")
load("bt_tune.rda")
# Linear Regression
lm_fit <- fit_resamples(lm_workflow, resamples = spotify_folds)
(lm_rmse <- show_best(lm_fit, metric = "rmse"))
# Ridge Regression
(ridge_rmse <- show_best(ridge_tune, metric = "rmse", n = 1))
# K-Nearest Neighbor
(knn_rmse <- show_best(knn_tune, metric = "rmse", n = 1))
# Elastic Net
(en_rmse <- show_best(en_tune, metric = "rmse", n = 1))
# Random Forest
(rf_rmse <- show_best(rf_tune, metric = "rmse", n = 1))
# Support Vector Machine
(svm_rmse <- show_best(svm_tune, metric = "rmse", n = 1))
# Boosted Trees
(bt_rmse <- show_best(bt_tune, metric = "rmse", n = 1))
Let’s find out which one of our models performed the best! We can find this by looking at the lowest RMSE value.
# Tibble comparing all of RMSE values
tibble_rmse <- tibble(Model = c("Linear Regression", "Ridge Regression", "K-Nearest Neighbors", "Elastic Net", "Random Forest", "Support Vector Machine", "Boosted Trees"), RMSE = c(lm_rmse$mean, ridge_rmse$mean, knn_rmse$mean, en_rmse$mean, rf_rmse$mean, svm_rmse$mean, bt_rmse$mean)) %>%
arrange(RMSE)
tibble_rmse
From this chart, we can tell that K-Nearest Neighbors performed the best because it had the lowest RMSE value. The next three best models were Random Forest, Ridge Regression, and Elastic Net.
Autoplots are a great way to see how each model performed and how tuning the parameters influenced the models. We can determine how well the models did for regression problems by looking at the RMSE. Low RMSE are indicated as better than models with high RMSE.
autoplot(knn_tune, metric = "rmse")
The K-Nearest Neighbors model was tuned to 5 levels. This plot shows
that with more nearest neighbors, the model does better. It has a pretty
high RMSE with 1 nearest neighbors and it decreases as more neighbors
are added. From this graph, we can see that the best would be with 15
neighbors.
autoplot(rf_tune, metric = "rmse")
The Random Forest model has
mtry, trees, and
min_n tuned and contains 8 levels. From the plot, it can be
seen that the number of trees do not have that much of an impact on the
model. Regardless of the number, they all follow an increasing line with
a slight curve. The minimal node size also does not have a huge impact
on the model since they also all have the same appearance. However, it
is noticeable that the number of randomly selected predictors does have
an influence on the RMSE. The less predictors we have, the lower the
RMSE.
autoplot(ridge_tune, metric = "rmse")
The Ridge Regression model has penalty tuned and we set mixture equal to 0. In this plot, we can see that the amount of regularization decreases pretty quickly and then increases again, ending with a stabilizing line.
Let’s take a look on what our best model is.
(knn_rmse <- show_best(knn_tune, metric = "rmse", n = 1))
Let’s give a Grammy to KNN Model 5 with 15 neighbors and a mean of 14.16711 which did the best out of all the models!
knitr::include_graphics("images/harrystyles.png")
Since KNN Model 5 is our best model, we will now use it to fit to our training data.
final_knn <- select_best(knn_tune, metric = "rmse")
final_knn_workflow <- finalize_workflow(knn_workflow, final_knn)
final_knn_fit <- fit(final_knn_workflow, data = spotify_train)
save(final_knn_fit, file = "final_knn_fit.rda")
load("final_knn_fit.rda")
Now that it has been fitted, we will use it on our testing data to see how well our model performed!
spotify_tibble <- predict(final_knn_fit, new_data = spotify_test
%>% select(-pop))
spotify_tibble <- bind_cols(spotify_tibble, spotify_test %>% select(pop))
spotify_tibble
final_knn_test <- augment(final_knn_fit, spotify_test)
rmse(final_knn_test, truth = pop, .pred)
Our model didn’t perform as well on our testing data as it did on our training but the RMSE scores were similar! To recall, the RMSE for our training set was 14.16711 and on our testing it is 15.12975. It is not the best but it did a decent job.
This is a scatterplot of the actual popularity values in the testing set versus the model-predicted values:
spotify_tibble %>%
ggplot(aes(x = .pred, y = pop)) +
geom_point(alpha = 0.5) +
geom_abline(lty = 2) +
coord_obs_pred() +
theme_minimal()
When we look at the plot, we can see that our model did not do the best job at predicting the popularity value but it did not do the worst either. A few points were able to be on the line which is good but the majority of the points are not. A couple reasons why the model is not insanely accurate could be that the data might be slightly imbalanced, even after doing the k-fold cross validation. If our testing data was imbalanced, this would lead to the KNN model being biased towards the majority value. KNN is also prone to overfitting or underfitting the data so this may also be the issue.
By exploring the data, building several machine learning models, and analyzing the outcomes, the best model to predict the popularity of Spotify songs is K-Nearest Neighbors. However, this model did not work perfectly and could be improved.
As for improvements, I should have considered including the
year and creating subsets of the data based on that. Music
is always changing and trends do change from year to year so the
attributes that would make a song more popular could vary. By including
the year, a stratified sample could be obtained so that there is a more
accurate representation of the commonalities between all the songs and
the years.
Another improvement I could have made would be creating a more elaborate recipe and having a more diverse dataset. With the majority of the dataset having the similar popularity scores, this may have created a biased outcome and made it more difficult to stratify. This would make it harder to train the model and producing an inaccurate prediction.
In general, my next step would be to continue working and hopefully implement the improvements I mentioned. I think it would be super cool and useful if I am able to lower my RMSE value and have a model that could get really close to predicting more popularity scores with recent data. Overall, this was a great opportunity to work on my machine learning skills and finding myself extremely passionate in creating more projects that involve machine learning. Hopefully, I can utilize the skills learned to make more projects and models that are worth a Grammy in my book!
knitr::include_graphics("images/beyonce.png")
The data set was called Top Spotify Songs From 2010-2019 By Year by Leonardo Henrique and it was retrieved from Kaggle.
Information in the introduction was from prior knowledge and from the Spotify site.